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Martin Sleziak
Martin Sleziak
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In$\DeclareMathOperator{\Spf}{Spf}\DeclareMathOperator{\Spa}{Spa}$In the Definition 8.5 of the paper "integral $p$ adic Hodge theory" by Bhatt-Morrow-Scholze, they define the adic generic fiber of a small affinoid smooth formal scheme $\textup{Spf}(R)$$\Spf(R)$ as $\textup{Spa}(R[\frac{1}{p}],R)$$\Spa(R[\frac{1}{p}],R)$, here $R$ is a smooth $\mathscr{O}_{\mathbb{C}_{p}}$-algebra which admits an 'etale map to the formal torus. However, the adic generic fiber defined by Huber should send $\textup{Spf}(R)$$\Spf(R)$ to $\textup{Spa}(R[\frac{1}{p}],R[\frac{1}{p}]^{\circ})$$\Spa(R[\frac{1}{p}],R[\frac{1}{p}]^{\circ})$, here $R[\frac{1}{p}]^{\circ}$ means the subring of all the power bounded elements of $R[\frac{1}{p}]$.

My question is that why $R=R[\frac{1}{p}]^{\circ}$ ? If $R$ is not smooth, a counterexample is $R=\mathscr{O}_{\mathbb{C}_{p}}\left \langle x,y \right \rangle/(xy-p)=\mathscr{O}_{\mathbb{C}_{p}}\left \langle x,\frac{p}{x} \right \rangle$(the semistable curve). In this example, $R[\frac{1}{p}]^{\circ}=\mathscr{O}_{\mathbb{C}_{p}}\left \langle x,y \right \rangle/(xy-1)=\mathscr{O}_{\mathbb{C}_{p}}\left \langle x,\frac{1}{x} \right \rangle$. Besides, if we consider the admissible blow up of the formal scheme $\textup{Spf}(R)$$\Spf(R)$, then the ring $R$ is changed, however, it does not change the generic fiber. Or my understanding about the adic generic fiber functor is wrong?

Any idea will be appreciated.

In the Definition 8.5 of the paper "integral $p$ adic Hodge theory" by Bhatt-Morrow-Scholze, they define the adic generic fiber of a small affinoid smooth formal scheme $\textup{Spf}(R)$ as $\textup{Spa}(R[\frac{1}{p}],R)$, here $R$ is a smooth $\mathscr{O}_{\mathbb{C}_{p}}$-algebra which admits an 'etale map to the formal torus. However, the adic generic fiber defined by Huber should send $\textup{Spf}(R)$ to $\textup{Spa}(R[\frac{1}{p}],R[\frac{1}{p}]^{\circ})$, here $R[\frac{1}{p}]^{\circ}$ means the subring of all the power bounded elements of $R[\frac{1}{p}]$.

My question is that why $R=R[\frac{1}{p}]^{\circ}$ ? If $R$ is not smooth, a counterexample is $R=\mathscr{O}_{\mathbb{C}_{p}}\left \langle x,y \right \rangle/(xy-p)=\mathscr{O}_{\mathbb{C}_{p}}\left \langle x,\frac{p}{x} \right \rangle$(the semistable curve). In this example, $R[\frac{1}{p}]^{\circ}=\mathscr{O}_{\mathbb{C}_{p}}\left \langle x,y \right \rangle/(xy-1)=\mathscr{O}_{\mathbb{C}_{p}}\left \langle x,\frac{1}{x} \right \rangle$. Besides, if we consider the admissible blow up of the formal scheme $\textup{Spf}(R)$, then the ring $R$ is changed, however, it does not change the generic fiber. Or my understanding about the adic generic fiber functor is wrong?

Any idea will be appreciated.

$\DeclareMathOperator{\Spf}{Spf}\DeclareMathOperator{\Spa}{Spa}$In the Definition 8.5 of the paper "integral $p$ adic Hodge theory" by Bhatt-Morrow-Scholze, they define the adic generic fiber of a small affinoid smooth formal scheme $\Spf(R)$ as $\Spa(R[\frac{1}{p}],R)$, here $R$ is a smooth $\mathscr{O}_{\mathbb{C}_{p}}$-algebra which admits an 'etale map to the formal torus. However, the adic generic fiber defined by Huber should send $\Spf(R)$ to $\Spa(R[\frac{1}{p}],R[\frac{1}{p}]^{\circ})$, here $R[\frac{1}{p}]^{\circ}$ means the subring of all the power bounded elements of $R[\frac{1}{p}]$.

My question is that why $R=R[\frac{1}{p}]^{\circ}$ ? If $R$ is not smooth, a counterexample is $R=\mathscr{O}_{\mathbb{C}_{p}}\left \langle x,y \right \rangle/(xy-p)=\mathscr{O}_{\mathbb{C}_{p}}\left \langle x,\frac{p}{x} \right \rangle$(the semistable curve). In this example, $R[\frac{1}{p}]^{\circ}=\mathscr{O}_{\mathbb{C}_{p}}\left \langle x,y \right \rangle/(xy-1)=\mathscr{O}_{\mathbb{C}_{p}}\left \langle x,\frac{1}{x} \right \rangle$. Besides, if we consider the admissible blow up of the formal scheme $\Spf(R)$, then the ring $R$ is changed, however, it does not change the generic fiber. Or my understanding about the adic generic fiber functor is wrong?

Any idea will be appreciated.

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user514790
user514790
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Adic generic fiber of a small formal scheme in the sense of Faltings

In the Definition 8.5 of the paper "integral $p$ adic Hodge theory" by Bhatt-Morrow-Scholze, they define the adic generic fiber of a small affinoid smooth formal scheme $\textup{Spf}(R)$ as $\textup{Spa}(R[\frac{1}{p}],R)$, here $R$ is a smooth $\mathscr{O}_{\mathbb{C}_{p}}$-algebra which admits an 'etale map to the formal torus. However, the adic generic fiber defined by Huber should send $\textup{Spf}(R)$ to $\textup{Spa}(R[\frac{1}{p}],R[\frac{1}{p}]^{\circ})$, here $R[\frac{1}{p}]^{\circ}$ means the subring of all the power bounded elements of $R[\frac{1}{p}]$.

My question is that why $R=R[\frac{1}{p}]^{\circ}$ ? If $R$ is not smooth, a counterexample is $R=\mathscr{O}_{\mathbb{C}_{p}}\left \langle x,y \right \rangle/(xy-p)=\mathscr{O}_{\mathbb{C}_{p}}\left \langle x,\frac{p}{x} \right \rangle$(the semistable curve). In this example, $R[\frac{1}{p}]^{\circ}=\mathscr{O}_{\mathbb{C}_{p}}\left \langle x,y \right \rangle/(xy-1)=\mathscr{O}_{\mathbb{C}_{p}}\left \langle x,\frac{1}{x} \right \rangle$. Besides, if we consider the admissible blow up of the formal scheme $\textup{Spf}(R)$, then the ring $R$ is changed, however, it does not change the generic fiber. Or my understanding about the adic generic fiber functor is wrong?

Any idea will be appreciated.